Github user cloud-fan commented on a diff in the pull request:

    https://github.com/apache/spark/pull/19479#discussion_r149058921
  
    --- Diff: 
sql/catalyst/src/main/scala/org/apache/spark/sql/catalyst/plans/logical/Statistics.scala
 ---
    @@ -275,6 +317,118 @@ object ColumnStat extends Logging {
           avgLen = row.getLong(4),
           maxLen = row.getLong(5)
         )
    +    if (row.isNullAt(6)) {
    +      cs
    +    } else {
    +      val ndvs = row.getArray(6).toLongArray()
    +      assert(percentiles.get.length == ndvs.length + 1)
    +      val endpoints = percentiles.get.map(_.toString.toDouble)
    +      // Construct equi-height histogram
    +      val buckets = ndvs.zipWithIndex.map { case (ndv, i) =>
    +        EquiHeightBucket(endpoints(i), endpoints(i + 1), ndv)
    +      }
    +      val nonNullRows = rowCount - cs.nullCount
    +      val ehHistogram = EquiHeightHistogram(nonNullRows.toDouble / 
ndvs.length, buckets)
    +      cs.copy(histogram = Some(ehHistogram))
    +    }
    +  }
    +
    +}
    +
    +/**
    + * There are a few types of histograms in state-of-the-art estimation 
methods. E.g. equi-width
    + * histogram, equi-height histogram, frequency histogram (value-frequency 
pairs) and hybrid
    + * histogram, etc.
    + * Currently in Spark, we support equi-height histogram since it is good 
at handling skew
    + * distribution, and also provides reasonable accuracy in other cases.
    + * We can add other histograms in the future, which will make estimation 
logic more complicated.
    + * This is because we will have to deal with computation between different 
types of histograms in
    + * some cases, e.g. for join columns.
    + */
    +trait Histogram
    +
    +/**
    + * Equi-height histogram represents column value distribution by a 
sequence of buckets. Each bucket
    + * has a value range and contains approximately the same number of rows.
    + * @param height number of rows in each bucket
    + * @param ehBuckets equi-height histogram buckets
    + */
    +case class EquiHeightHistogram(height: Double, ehBuckets: 
Array[EquiHeightBucket])
    +  extends Histogram {
    +
    +  // Only for histogram equality test.
    +  override def equals(other: Any): Boolean = other match {
    +    case otherEHH: EquiHeightHistogram =>
    +      height == otherEHH.height && 
ehBuckets.sameElements(otherEHH.ehBuckets)
    +    case _ => false
    +  }
    +
    +  override def hashCode(): Int = super.hashCode()
    +}
    +
    +/**
    + * A bucket in an equi-height histogram. We use double type for 
lower/higher bound for simplicity.
    + * @param lo lower bound of the value range in this bucket
    + * @param hi higher bound of the value range in this bucket
    + * @param ndv approximate number of distinct values in this bucket
    + */
    +case class EquiHeightBucket(lo: Double, hi: Double, ndv: Long)
    +
    +object HistogramSerializer {
    +  // A flag to indicate the type of histogram
    +  val EQUI_HEIGHT_HISTOGRAM_TYPE: Byte = 1
    +
    +  /**
    +   * Serializes a given histogram to a string. For advanced statistics 
like histograms, sketches,
    +   * etc, we don't provide readability for their serialized formats in 
metastore (as
    +   * string-to-string table properties). This is because it's hard or 
unnatural for these
    +   * statistics to be human readable. For example, histogram is probably 
split into multiple
    +   * key-value properties, instead of a single, self-described property. 
And for
    +   * count-min-sketch, it's essentially unnatural to make it a readable 
string.
    +   */
    +  final def serialize(histogram: Histogram): String = histogram match {
    +    case h: EquiHeightHistogram =>
    +      // type + numBuckets + height + numBuckets * (lo + hi + ndv)
    --- End diff --
    
    what's the common size of `numBuckets`? If it's large enough, we may need 
to consider compression.


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